Nonlinear Equations for Finite-Amplitude Wave Propagation in Fiber-Reinforced Hyperelastic Media
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1 Nonlinear Equations for Finite-Amplitude Wave Propagation in Fiber-Reinforced Hyperelastic Media Alexei F. Cheviakov Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, Canada December 7, 2014 A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
2 Collaborators J.-F. Ganghoffer, LEMTA - ENSEM, Université de Lorraine, Nancy, France Simon St. Jean, M.Sc. graduate, University of Saskatchewan A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
3 Notation Notation u x ux. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
4 Examples Collagen fiber in connective biological tissue (cnx.org/content/m46049/latest/) A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
5 Examples Arterial tissue (Holzapfel, Gasser, and Ogden, 2000) A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
6 Examples Fabric A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
7 Examples Appropriate framework: incompressible hyperelasticity. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
8 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). A solid body occupies the reference (Lagrangian) volume Ω 0 R 3. in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Actual (Eulerian) configuration: Ω R oordinates. The deformed body occupies an 3. Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given Material by points are labeled by X Ω 0. = dx dt The dφ actual dt. position of a material point: x = x (X, t) Ω. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
9 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). in the Velocity reference ofconfiguration a material point arex: commonly v (X, t) = dx referred dt. to as Lagrangian coordinates, and ac oordinates. Jacobian Thematrix deformed (deformation body occupies gradient): an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given by = dx dt dφ dt. F(X, t) = φ; J = det F > 0; g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
10 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). Boundary force (per unit area) in Eulerian configuration: t = σn. in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Boundary force (per unit area) in Lagrangian configuration: T = PN. oordinates. The deformed body occupies an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given σ = by σ(x, t) is the Cauchy stress tensor. = dx dt P = dφ JσF T dt. is the first Piola-Kirchhoff tensor. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
11 Notation; Material Picture A.F. Cheviakov, J.-F. Ganghoffer / J. Math. Anal. Appl. 396 (2012) Fig. 1. Material and Eulerian coordinates. osition x of a material point labeled by X Ω 0 at time t is given by Material picture, t), x i = φ i (X, t). Density in reference configuration: ρ 0 = ρ 0(X) (time-independent). in the reference configuration are commonly referred to as Lagrangian coordinates, and ac Density in actual configuration: oordinates. The deformed body occupies an Eulerian domain Ω = φ(ω 0 ) R 3 (Fig. 1). T X is given by ρ = ρ(x, t) = ρ 0/J. = dx dt dφ dt. g φ must be sufficiently smooth (the regularity conditions depending on the particular proble A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
12 Governing Equations Variables: Independent: time t, Lagrangian coordinates X Ω 0. Dependent: x = x(x, t), p = p(x, t), ρ = ρ(x, t). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
13 Governing Equations Variables: Independent: time t, Lagrangian coordinates X Ω 0. Dependent: x = x(x, t), p = p(x, t), ρ = ρ(x, t). Incompressibility: J = det F = x i X j = 1, ρ = ρ0/j = ρ0(x). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
14 Governing Equations Variables: Independent: time t, Lagrangian coordinates X Ω 0. Dependent: x = x(x, t), p = p(x, t), ρ = ρ(x, t). Incompressibility: J = det F = x i X j = 1, ρ = ρ0/j = ρ0(x). Equations of motion: ρ 0x tt = div (X ) P + ρ 0R, J = 1. R = R(X, t): total body force per unit mass. (div (X ) P) i = Pij X j. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
15 Constitutive Relations The first Piola-Kirchhoff stress tensor (incompressible): P ij = p (F 1 ) ji + ρ 0 W F ij, (1) W = W (X, F): a scalar strain energy density; p = p (X, t): hydrostatic pressure. Strain Energy Density W = W iso + W aniso. Isotropic Strain Energy Density For the left Cauchy-Green strain tensor B = FF T, I 1 = Tr B = F i kf i k, I 2 = 1 [(Tr 2 B)2 Tr(B 2 )], I 3 = det B = J 2 = 1. (2) Mooney-Rivlin materials: W iso = a(i 1 3) + b(i 2 3), a, b > 0, A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
16 Constitutive Relations Anisotropic Strain Energy Density In the reference configuration, fibers are oriented along the unit vector A = A(X). Actual fiber direction: a = a(x, t) = FA/ FA = FA/λ; λ = FA is the fiber stretch factor. Fiber invariants: I 4 = A T CA, I 5 = A T C 2 A. General model: W aniso = f (I 4 1, I 5 1), f (0, 0) = 0. Various specific models are used. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
17 Full Set of Equations Equations of motion: ρ 0x tt = div (X ) P, [ ] x i J = det = 1, X j P ij = p (F 1 ) ji + ρ 0 W F ij. 4 PDEs, 4 unknowns. Specific constitutive relation: W = W iso + W aniso = a(i 1 3) + b(i 2 3) + q (I 4 1) 2 ; a, b, q > 0. Isotropic part: Mooney-Rivlin-type; Anisotropic part: quadratic reinforcing model. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
18 Ansatz 1 Compatible with Incompressibility Equilibrium and Displacements Equilibrium/no displacement: x = X, natural state. Time-dependent, with displacement: x = X + G, G = G(X, t). No linearization/assumption of smallness of G. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
19 Ansatz 1 Compatible with Incompressibility Equilibrium and Displacements Equilibrium/no displacement: x = X, natural state. Time-dependent, with displacement: x = X + G, G = G(X, t). No linearization/assumption of smallness of G. Motions Transverse to a Plane x = X 1 X 2 X 3 + G ( X 1, X 2, t ), A = cos γ 0 sin γ. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
20 Ansatz 1 Compatible with Incompressibility Equilibrium and Displacements Equilibrium/no displacement: x = X, natural state. Time-dependent, with displacement: x = X + G, No linearization/assumption of smallness of G. Motions Transverse to a Plane x = X 1 X 2 X 3 + G ( X 1, X 2, t ), A = G = G(X, t). cos γ 0 sin γ. Deformation gradient: F = G/ X 1 G/ X 2 1, J = F 1. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
21 F = , G/ X 1 G/ X 2 1 Ansatz 1 Compatible with Incompressibility one has J 1, and the incompressibility condition is identically satisfied. X 3 A X 2 X 1 (a) 1 1 X 3 0 x X X 1-1 x x 1-1 (b) (c) (a) Fiber direction. Figure 1: (b) (a) Fiber Thedirection. reference (b)(lagrangian) The reference (Lagrangian) mesh with mesh fibers. with fibers. (c) A(c) sample A sample deformed mesh (Eulerian deformed configuration) mesh (Eulerian for configuration) this ansatz. for the ansatz (4.1). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
22 Ansatz 1 Compatible with Incompressibility Current model Unknowns: G ( X 1, X 2, t ) and p ( X 1, X 2, t ). Parameters: Isotropic part: a, b = const > 0. Anisotropic part: q = const > 0. Fiber angle: γ. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
23 Ansatz 1 Compatible with Incompressibility Equations of motion: ( ) 2 G 2 G = 2 (a + b) 2 t (X 1 ) + 2 G ( 2 (X 2 ) 2 ( ) 2 G +4q cos 2 (γ) 3 cos 2 (γ) + 6 cos(γ) sin(γ) G X 1 X sin2 γ 0 = p ( G X + 1 2bρ0 X 1 2 G 8qρ 0 cos 3 γ (X 1 ) 2 0 = p ( G X + 2 2bρ0 X 2 2 G (X 2 ) G ) 2 G ( 2 X 2 X 1 X 2 cos γ G ) X + sin γ, 1 2 G (X 1 ) G 2 X 1 ) 2 G. X 1 X 2 ) 2 G (X 1 ) 2, A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
24 Ansatz 1 Compatible with Incompressibility Nonlinear wave equation with a differential constraint: Wave equation: ( ) 2 G 2 G = 2 (a + b) 2 t (X 1 ) + 2 G ( 2 (X 2 ) 2 ( ) 2 G +4q cos 2 (γ) 3 cos 2 (γ) + 6 cos(γ) sin(γ) G X 1 X sin2 γ Differential constraint: b [ G 2 G X 1 X 1 X 1 X G 2 X 2 = [ 4q cos 3 γ 2 G X 2 (X 1 ) 2 ] 2 G (X 1 ) 2 ( cos γ G X 1 + sin γ ) ( G + b X 2 ) 2 G X 1 X G 2 X 1 2 G (X 1 ) 2, )] 2 G (X 2 ) 2 A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
25 Ansatz 1 Compatible with Incompressibility Special case 1: fibers in X 3 -direction b X 1 [ G ( 2 G X 2 (X 1 ) G = 2 (a + b) 2 t ( 2 G (X 1 ) 2 + )] 2 G + b (X 2 ) 2 X 2 [ G X 1 2 G (X 2 ) 2 ), ( 2 G (X 1 ) 2 + )] 2 G = 0, (X 2 ) 2 No fiber effect. Here G = G(X 1, X 2, t). General fact: if the displacement does not depend on the fiber direction, i.e., ( G)A = 0, then I 4 J = const. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
26 Ansatz 1 Compatible with Incompressibility Special case 2: one-dimensional displacements X = X 1 X 2 X 3 + G ( X 1, t ) ; then (denote X 1 = x, G = G(x, t)) G tt = ( ( )) α + β cos 2 γ 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, (no constraint). Pressure is given by p = βρ 0 cos 3 γ (cos γg x + 2 sin γ) G x + f (t). Here α = 2(a + b) > 0, β = 4q > 0. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
27 One Dimensional Transverse Waves Reference Configuration Actual Configuration A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
28 Symmetries of the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Parameters arbitrary Symmetries Z 1 = x, Z2 = t, Z3 = G, Z4 = t G, Z 5 = x x + t t + G G 4α β, Z 1, Z 2, Z 3, Z 4, Z 5, ( ) cos 2 γ = 1 1 ± 1 4α Z 6 = 2t cos γ 2 β t + x cos γ x x sin γ G A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
29 Symmetries of the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Parameters arbitrary Symmetries Z 1 = x, Z2 = t, Z3 = G, Z4 = t G, Z 5 = x x + t t + G G 4α β, Z 1, Z 2, Z 3, Z 4, Z 5, ( ) cos 2 γ = 1 1 ± 1 4α Z 6 = 2t cos γ 2 β t + x cos γ x x sin γ G The extra symmetry Z 6 can arise only for materials where the fiber contribution is sufficiently strong: β > 4α q > 2(a + b) (W = a(i 1 3) + b(i 2 3) + q (I 4 1) 2 ). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
30 Remarks on the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Remarks: Wave equations of the form u tt = F (u x)u xx can be linearized by a Hodograph transformation. Can be mapped by a point transformation into a constant-coefficient equation for a restricted class of F (u x). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
31 Remarks on the Nonlinear Wave Equation PDE on G(x, t): G tt = ( α + β cos 2 γ ( )) 3 cos 2 γ (G x) sin γ cos γg x + 2 sin 2 γ G xx, Remarks: Wave equations of the form u tt = F (u x)u xx can be linearized by a Hodograph transformation. Can be mapped by a point transformation into a constant-coefficient equation for a restricted class of F (u x). PDE when γ = 0: transverse 1D waves propagating along the fibers ( G tt = α + 3β (G x) 2) G xx. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
32 1D Displacements Orthogonal to the Fibers Numerical solution G x p x A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
33 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
34 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. Deformation gradient: F = H/ X G/ X 1 0 1, J = F 1. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
35 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. Deformation gradient: F = H/ X G/ X 1 0 1, J = F 1. Governing PDEs: Denote X 1 = x, G = G(x, t), H = H(x, t). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
36 Ansatz 2 Compatible with Incompressibility Displacements transverse to an axis: X = X 1 X 2 + H ( X 1, t ) X 3 + G ( X 1, t ), A = cos γ 0 sin γ. Coupled nonlinear wave equations: 0 = p x 2βρ 0 cos 3 γ [(cos γg x + sin γ) G xx + cos γh xh xx], [ H tt = αh xx + β cos 3 γ cos γ ([ ] ] ) Gx 2 + Hx 2 Hxx + 2G xh xg xx + 2 sin γ x (GxHx), G tt = αg xx + β cos 2 γ [ 2 sin 2 γ G xx + cos 2 γ ( 2G xh xh xx + ( ) ) Hx 2 + 3Gx 2 Gxx + sin 2γ (3G xg xx + H xh xx)]. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
37 Ansatz 2 Compatible with Incompressibility Case 1: γ = π/2 H tt = αh xx, G tt = αg xx. Case 2: γ = 0 H tt = αh xx + β [([ 3H 2 x + G 2 x ] Hxx + 2G xh xg xx )], G tt = αg xx + β [( 2G xh xh xx + ( H 2 x + 3G 2 x ) Gxx )]. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
38 Ansatz 2 Compatible with Incompressibility Case 1: γ = π/2 H tt = αh xx, G tt = αg xx. Case 2: γ = 0 H tt = αh xx + β [([ 3H 2 x + G 2 x ] Hxx + 2G xh xg xx )], G tt = αg xx + β [( 2G xh xh xx + ( H 2 x + 3G 2 x ) Gxx )]. Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
39 Motions Transverse to an Axis: Traveling Wave Solutions Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
40 Motions Transverse to an Axis: Traveling Wave Solutions Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. Solutions: (g ) 2 + (h ) 2 = R 2 = c2 α, when c > c 0 = α = 2(a + b). β A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
41 Motions Transverse to an Axis: Traveling Wave Solutions Traveling-wave solutions when γ = 0 r = x ct, H(x, t) = h(r), G(x, t) = g(r); [ α c 2 + β(3(h ) 2 + (g ) 2 ) ] h + 2βg h g = 0, 2βg h h + [ α c 2 + β((h ) 2 + 3(g ) 2 ) ] g = 0. Solutions: (g ) 2 + (h ) 2 = R 2 = c2 α, when c > c 0 = α = 2(a + b). β Sample solution: a traveling periodic wave h(r) = A cos(kr + φ 0), g(r) = A sin(kr + φ 0), A = R/k. A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
42 Sample solution: a traveling periodic wave An exact solution: h(r) = A cos(kr + φ 0), g(r) = A sin(kr + φ 0), A = R/k. This describes a time-periodic perturbation of the stress-free steady state, given by X 1 0 x = x (X, t) = X 2 + A cos(k[x 1 ct] + φ 0). X 3 Every material point follows a circle. sin(k[x 1 ct] + φ 0) Material lines along the X 1 direction, given by X 2 = const and X 3 = const, become helices parameterized by X 1 (figure below). A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
43 Sample solution: a traveling periodic wave (a) (b) AFigure time-periodic 5: Sometraveling material perturbation. lines for XMaterial 2 = const, lines Xalong 3 = const the Xin 1 direction, the reference givenconfiguration by (a XThe 2, Xsame 3 = const, lines are in the shown. actual configuration, parameterized by (5.22) (b). 6 Conclusions A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
44 Conclusions and Open Problems Current work A class of models of anisotropic hyperelastic fiber-reinforced materials is considered. Nonlinear wave equations are derived for ansätze compatible with incompressibility. Symmetry properties and exact solutions are being studied. Future/ongoing work Work with two fiber families. Consider setups more closely related to applications: A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
45 Some references Ciarlet, P. G. Mathematical Elasticity. Volume I: Three-dimensional Elasticity. Elsevier, Marsden, J. E. and Hughes, T. J. R. Mathematical Foundations of Elasticity. Dover, C.A. Basciano, C. Kleinstreuer. Invariants-based anisotropic constitutive models of healthy aneurysmal abdominal aortic wall. J. Biomech. Eng., 131:021009, G.A. Holzapfel, T.C. Gasser, R.W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity, 61:1 48, Cheviakov, A.F., and Ganghoffer, J.-F. Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics. J. Math. An. App., A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
46 Some references Ciarlet, P. G. Mathematical Elasticity. Volume I: Three-dimensional Elasticity. Elsevier, Marsden, J. E. and Hughes, T. J. R. Mathematical Foundations of Elasticity. Dover, C.A. Basciano, C. Kleinstreuer. Invariants-based anisotropic constitutive models of healthy aneurysmal abdominal aortic wall. J. Biomech. Eng., 131:021009, G.A. Holzapfel, T.C. Gasser, R.W. Ogden. A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity, 61:1 48, Cheviakov, A.F., and Ganghoffer, J.-F. Symmetry Properties of Two-Dimensional Ciarlet-Mooney-Rivlin Constitutive Models in Nonlinear Elastodynamics. J. Math. An. App., Thank you for your attention! A. F. Cheviakov (U. Saskatchewan) Wave Models in Fiber-Reinforced Hyperelasticity CMS Winter mtg / 31
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